(A): Both transverse and longitudinal mechanical waves can travel through solids, whereas only longitudinal mechanical waves can travel through gases.
(R): Gases do not have rigidity modulus.

Options

A. Both 'A' and 'R' are true and 'R' is the correct, explanation of 'A'.
B. Both 'A' and 'R' are true and 'R' is not the correct explanation of 'A'
C. A' is true and 'R' is false
D. Both 'A' and 'R' are false

Answer

To analyze the given statements (A) and (R), let's break down the concepts involved step by step.

### Step 1: Understanding Mechanical Waves
Mechanical waves are disturbances that propagate through a medium due to the interaction of particles. There are two main types of mechanical waves: transverse and longitudinal.

### Step 2: Characteristics of Transverse Waves
Transverse waves are waves where the particle displacement is perpendicular to the direction of wave propagation. For transverse waves to propagate, the medium must possess rigidity, which is characterized by the shear modulus.

### Step 3: Characteristics of Longitudinal Waves
Longitudinal waves are waves where the particle displacement is parallel to the direction of wave propagation. These waves can travel through all states of matter: solids, liquids, and gases. Longitudinal waves require the medium to have bulk modulus, which is a measure of how incompressible a substance is.

### Step 4: Behavior of Waves in Solids and Gases
- **In Solids**: Both transverse and longitudinal waves can travel because solids possess both shear modulus (for transverse waves) and bulk modulus (for longitudinal waves).
- **In Gases**: Only longitudinal waves can travel because gases do not have shear modulus (rigidity) required for transverse waves.

### Step 5: Evaluating Statement (A)
Statement (A) claims that both transverse and longitudinal mechanical waves can travel through solids, while only longitudinal mechanical waves can travel through gases. This statement is true based on the properties of solids and gases discussed above.

### Step 6: Evaluating Statement (R)
Statement (R) states that gases do not have rigidity modulus. This is also true because gases lack the structural rigidity required to support shear forces, which is necessary for the propagation of transverse waves.

### Conclusion
Since both statements (A) and (R) are true, and (R) correctly explains the reason behind (A), we conclude that both statements are correct, and (R) is a valid reason for (A).

### Final Answer
Both statements are true: (A) is true, and (R) is true because gases do not have rigidity modulus.

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